I have an image of a 2D-sinusoidal pattern $f(x, y)$ with wavelength $\lambda$ which I would like to convolve with a 2D circular pill-box function $h(x,y)$ of radius $r$.
The image is given by $$ f(x,y) = \dfrac{1}{2}\bigg[1 + \sin\bigg(\dfrac{2\pi x}{\lambda}\bigg)\bigg]. $$ Similarly the circular pill-box function is given by $$ h(x,y)= \dfrac{1}{\pi r^2} \begin{cases} 1& \text{if } x^2 + y^2 \leq r^2\\ 0 & \text{otherwise} \end{cases} $$ $\qquad$ where the scaling constant $\dfrac{1}{\pi r^2}$ ensures that the area of the filter is one.
The 2D convolution integral may be expressed as $$ g(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\tau_u, \tau_v)\, h(x - \tau_u, y-\tau_v) \, \mathrm{d}\tau_u \, \mathrm{d}\tau_v $$
$\qquad$ where $\tau$ is a dummy variable to represent the shift of one function with respect to the other.
How do I evaluate the convolution integral so that I can express the convolved image $g(x,y)$ in analytic form.
Thanks in advance : )